A statistical resolution measure of fluorescence microscopy with finite photons

First discovered by Ernest Abbe in 1873, the resolution limit of a far-field microscope is considered determined by the numerical aperture and wavelength of light, approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\lambda }{2{NA}}$$\end{document}λ2NA. With the advent of modern fluorescence microscopy and nanoscopy methods over the last century, this definition is insufficient to fully describe a microscope’s resolving power. To determine the practical resolution limit of a fluorescence microscope, photon noise remains one essential factor yet to be incorporated in a statistics-based theoretical framework. We proposed an information density measure quantifying the theoretical resolving power of a fluorescence microscope in the condition of finite photons. The developed approach not only allows us to quantify the practical resolution limit of various fluorescence and super-resolution microscopy modalities but also offers the potential to predict the achievable resolution of a microscopy design under different photon levels.


Supplementary Text
Note 1: Resolution criterion threshold on information density The information-based resolution (IbR) is defined as the reciprocal of frequency where information density   drops below 10 rad -2 μm -2 threshold.For an object of  =  λ , applying threshold   = 10  −2  −2 to a unit area (one cycle of the sinusoidal wave), the Fisher information corresponds to phase estimation uncertainty   =  2.2 (Fig. S2).Since Combining first equation we have Last step of inequality comes from fact that any Fisher information is greater or equal to 0. If the processing is deterministic, then Fisher information in the processed data  would always be smaller or equal to raw data .In another word, deterministic image post-processing cannot increase Fisher information of any parameter.Note that this statement is under condition that image processing is deterministic.If other prior information could be accessed, this relationship would not stand.

Note 3: Pixel binning effect on image
The pixel binning effect is well known for functioning as low pass filter (33).We derive the pixel binning effect in mathematical formula as follows.
Supposing . Then Applying Fourier transform on function (, ) we have Let  ′ =  + ,  ′ =  + , we have Exchange the integration sequence we have From expression, function (, ) is function (, ) frequency filtered by two sinc functions.
The image pixels' value [, ] is the sampling on function (  ,   ).Pixel size  not only affects the sampling process but also affects the frequency filter applied on the image.
OTF and low pass filter by pixelization in 1d.When the pixel size is at the Nyquist sampling requirement: , the transmission rate at  = where (, ) is the fluorophore density distribution,   (, ) is the excitation PSF, ( 0 ,  0 ) denotes the scanning position and  is the excitation to emission coefficient we assumed to be 1 for simplicity.
We also assumed the magnification is 1 for simplicity.
In confocal imaging, a bucket photon detector would collect all photons for each scanning position and form a value.Therefore, the ideal image value of confocal at ( 0 ,  0 ) would be, Exchange the integration order we have, If we assume the pinhole function is symmetric, we have, Let  ′ =  −  and  ′ =  − , the equation becomes, The integration in the bracket above is considered as a convolution between  and  ℎ taking the value at ( 0 + ,  0 + ), then the equation becomes, Consider the conjugate relationship ( 0 ,  0 ) = (− 0 , − 0 ) and we have, Expand the fluorescence distribution expression we have, If we consider  and   are also symmetric, the equation becomes, Writing in this expression allows us to identify the ideal image formation in confocal as object convolution with an effective PSF where, The corresponding effective OTF would be Fourier transform on the PSF expression, Note 5: Conceptual differences between IbR and FRC Information based resolution (IbR) is different from any Fourier Ring Correlation (FRC) defined resolution method, where FRC is directly computed from an observed image (13,14).In contrast, our Fisher information density is a theoretical measure based on the imaging conditions, independent of specific observed image.The noisy images we added to the figures in manuscript are simulated noisy images for visualization only.They are not used for our Fisher information calculation.Fisher information is purely based on deterministic theoretical calculation and thus the information density or IbR can be considered as a function of system and sample parameters.

Figures of effective OTF
() = information(, , , ℎ, , modality, , bg) Where  is refractive index,  is the numerical aperture,  is the emission wavelength, ℎ is the emitted photons from target object,  is the frequency of the object, modality represents a particular microscopy design,  is the volume thickness, bg is the background fluorescence density.In contrast, FRC is a post-acquisition resolution evaluation on a specific noisy image.

𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐹𝑅𝐶(acquired_image)
The meaning of our theoretical Fisher information measure for resolving power is that it provides a potential way to compare different imaging modalities theoretically in a statistical manner.In Abbe's resolution consideration, resolution is only affected by numerical aperture NA and wavelength λ.Our information density allows us to rewrite resolution as a function of the emitted photons. .SIM uses 9 illumination patterns with 3 illumination orientations each having 3 phase patterns.One of SIM's illumination patterns aligned with the sine pattern object.Camera pixel size in the wide-field system and SIM equal scanning intervals in the confocal system and ISM and they were set to 0.1 μm (0.16 AU).Photon collection efficiencies were considered based on 4Pi solid angle emission, objective NA and the pinhole rejection.The above simulation had photon collecting efficiency 32.05% for wide-field and SIM microscopy, 10.77% for confocal microscopy with a 0.5 AU pinhole, and 24.68% for ISM with a 1.3 AU FOV.Supplementary Fig. 7. Fisher information calculation in four imaging modalities with respect to detected photons and frequency.Red planes indicated the resolving criterion   = 10  −2 •  −2 .The simulation conditions were conducted in planar specimen (background free), numerical aperture of 1.4, immersion medium refractive index of 1.5, and emission wavelength of 0.7 μm.The confocal system is configured with a pinhole diameter of 0.5 AU.ISM modality is set with a detector pixel size of 0.26 AU with 5 by 5 pixels, covering a 1.3 AU square.SIM employs a structured illumination frequency of   = 2  , with nine illumination patterns in 3 illumination orientations, one aligning with the sinusoidal pattern object.Each illumination orientation had three phase patterns.The camera pixel size for widefield system and SIM, as well as the scanning intervals for confocal and ISM, were set to 0.1 μm (0.16 AU).
637 (red curve).Note 4: Effective OTF of confocal Diagram of a confocal microscope.The OTF derivation of confocal can be found in works (23,40).For completeness and reference, we provide our version of derivation below.The fluorescence distribution on the object plane is   (, ) = (, ) •   ( −  0 ,  −  0 ) • , Figures of effective OTF of confocal with different pinhole diameters are plotted in Fig. S11.

Note 6 :,
photons/μm 2 , background photon emission density 500 photons/μm 3 , numerical aperture of 1.4, immersion medium refractive index of 1.5, and emission wavelength of 0.7 μm.Confocal system considered a pinhole diameter of 0.5 AU.ISM considered a detector pixel size 0.26 AU with 5 by 5 pixels, covering detector area 1.3 AU × 1.3 AU.SIM implemented a structured illumination frequency of   = 2